A247304
Expected value of trace(O)^(2n), where O is a 5 X 5 orthogonal matrix randomly selected according to Haar measure.
Original entry on oeis.org
1, 3, 15, 105, 945, 10263, 127699, 1751685, 25807445, 401449271, 6519160851, 109656966853, 1899821492925, 33757439931675, 613135175204151, 11352879180474205, 213825919244349885, 4089023317563827335, 79269914519539177315, 1555788699527063688453
Offset: 1
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A247304 := proc (n) return (1/2)*integrate(integrate((1-cos(x))*(1-cos(y))*(cos(x)-cos(y))^2*(1+2*cos(x)+2*cos(y))^(2*n), y = 0 .. 2*Pi), x = 0 .. 2*Pi)/Pi^2 end proc; seq(A247304(n), n = 1 .. 20);
A138356
Moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(4).
Original entry on oeis.org
1, 1, 2, 4, 10, 27, 82, 268, 940, 3476, 13448, 53968, 223412, 949535, 4128594, 18310972, 82645012, 378851428, 1760998280, 8288679056, 39457907128, 189784872428, 921472827272, 4512940614960, 22279014978544, 110797225212112
Offset: 0
a(3) = 4 because E[X^3] = 4 for X the t^2 coeff of det(tI-A) in USp(4).
a(3) = 1*2^3*(1*1-0^2) + 3*2^2*(0*0-1^2) + 3*2^1*(1*2-0^2) + 1*2^0*(0*0-2^2) = 8 - 12 + 12 - 4 = 4.
- Kiran S. Kedlaya, Andrew V. Sutherland, Computing L-series of hyperelliptic curves, arXiv:0801.2778 [math.NT], 2008-2012; Algorithmic Number Theory Symposium--ANTS VIII, 2008.
- Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
- Nicholas M. Katz and Peter Sarnak, Random Matrices, Frobenius Eigenvalues and Monodromy, AMS, 1999.
A247591
Dimension of invariants of 2n-th tensor power of 6-dimensional irreducible representation of A_3.
Original entry on oeis.org
1, 1, 3, 16, 126, 1296, 16071, 228514, 3607890, 61891050, 1135871490, 22049362440, 448790912004, 9512960347260, 208858963314735, 4728736078065810, 110006925920592810, 2621619942885055530, 63840054782606886630, 1585094577104979776880, 40054740803371374834780, 1028483346608802276173280
Offset: 0
For 2n=6, there are 15 invariants corresponding to all ways of pairing the 6 indices with the metric tensor, plus one invariant which is the completely skew-symmetric 6-index tensor.
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a[0] = 1; a[1] = 1; a[n_] := a[n] = (4*n*(2*n-1)*(5*n+7)*a[n-1] - 36*(n-1)*(2*n-3)*(2*n-1)*a[n-2]) / ((n+2)*(n+3)^2); Table[a[n], {n, 0, 21}]
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N=66; v=vector(N); v[1]=1; v[2]=1;
for(n=2, N-1, my(t=n+1); v[t] = (-36*(n-1)*(2*n-3)*(2*n-1)*v[t-2] + 4*n*(2*n-1)*(5*n+7)*v[t-1]) / ((n+2)*(n+3)^2) );
v \\ Joerg Arndt, Sep 20 2014
A353173
Dimension of space of invariants of n-th tensor power of the 26-dimensional fundamental (or "standard") irreducible representation of F_4.
Original entry on oeis.org
1, 0, 1, 1, 5, 15, 70, 330, 1820, 10858, 70875, 497135, 3727955, 29658410, 248989676, 2194891440, 20231692430, 194286848280, 1937546532820, 20008993160460, 213436182918652, 2346406693816315, 26531060178217182, 307987244037724262, 3664579007885995952
Offset: 0
a(1)=0 because there is no F_4-invariant linear form on the 26-dimensional representation; a(2)=1 because there is, up to scalars, precisely one invariant quadratic form.
The analogous sequence for the (52-dimensional) adjoint representation of F_4 is:
A179685.
A similar sequence for G_2 (for its 7-dimensional fundamental irreducible representation) is:
A059710.
A similar sequence for B_2 (for its standard 5-dimensional irreducible representation) is:
A095922.
For A_n the similar sequence (omitting some 0's) is given by the (n+1)-dimensional Catalan numbers, e.g.,
A005789 for A_2.
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p_tensor(n,[0,0,0,1],F4)|[0,0,0,0] # Returns the value of a(n).
Showing 1-4 of 4 results.
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